Construction of a Generalized Solution of a Mixed Problem for the Telegraph Equation: Sequential and Axiomatic Approaches

Under minimal conditions on the initial data of the mixed problem for the telegraph equation, a generalized solution is obtained in two different ways in the form of a rapidly converging function series—an analog of the well-known d’Alembert formula. The first approach is based on the sequential method: a generalized solution of the problem is determined as the limit of classical solutions of a sequence of problems. The second approach is based on the axiomatic method. Classical solutions are not involved in constructing a generalized solution. Euler’s theory of divergent series with an augmented system of axioms is used. The specific feature of the problem under consideration is the presence of a nonlocal boundary condition in which the value of the function at an interior point of the interval occurs. Both approaches to constructing a generalized solution lead to the same rapidly (exponentially) converging function series.